INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith

Enter your Sign on user name and password.

Forgot password?
  • Follow us on:
Loading video...
Text Comments (3)

2 answers

Last reply by: Professor Eric Smith
Mon Dec 2, 2013 8:41 PM

Post by Mirza Baig on November 23, 2013

Why do we always write as square like x^2+16 as x^2+ 4^2  Why ?????

Special Factoring Techniques

  • By memorizing some special polynomials, you can have quick formulas for factoring. These include
    • The difference of squares
    • Perfect square trinomials
    • The sum and difference of cubes
  • Make sure to factor out a greatest common factor before using any of these special formulas.
  • Note that there is not a special formula for the sum of squares.
  • When using the formula for the sum or difference of cubes, note how the signs follow SOP. This stands for Same Opposite Positive.
  • Some of these special formulas could be used more than once in a particular problem.

Special Factoring Techniques

Factor:
25x2 − 64

  • 25x2 = a2
  • a = 5x
  • 64 = b2
  • b = 8

( 5x + 8 )( 5x − 8 )

Factor:
81y2 − 121

  • a2 = 81y2
  • a = 9y
  • b2 = 121
  • b = 11

( 9y + 11 )( 9y − 11 )

Factor:
49t2 − 225

  • a2 = 49t2
  • a = 7t
  • b2 = 15

( 7t + 15 )( 7t − 15 )

Factor:
3m4 − 27m2

  • 3m2( m2 − 9 )
  • a2 = m2
  • a = m
  • b2 = 9
  • b = 3

3m2( m + 3 )( m − 3 )

Factor:
72k4 − 288k2

  • 2k2( 36k2 − 144 )
  • a2 = 36k2
  • a = 6k
  • b2 = 144
  • b = 12

2k2( 6k + 12 )( 6k − 12 )

Factor:
12p4 − 75p2

  • 3p2( 4p2 − 25 )
  • a2 = 4p2
  • a = 2p
  • b2 = 25

b = 5

Factor:
6n4 − 18n3 − 10n2 + 30n

  • 2n(3n3 − 9n2 − 5n + 15)
  • 2n[ ( 3n3 − 9n ) + ( − 5n + 15 ) ]
  • 2n[ 3n2( n − 3 ) − 5( n − 3 ) ]

2n[ ( 3n2 − 5 )( n − 3 ) ]

Factor:
48x4 + 96x3 − 75x2 − 150x

  • 3x( 16x3 + 32x2 − 25x − 50 )
  • 3x[ ( 16x3 + 32x2 ) + ( − 25x − 50 ) ]
  • 3x[ 16x3( x + 2 ) − 25( x + 2 ) ]
  • 3x[ ( 16x2 − 25 )( x + 2 ) ]

3x[ ( 4x + 5 )( 4x − 5 )( x + 2 ) ]

Factor:
5y4 − 34y3 − 20y2 − 35y

  • 5y( y3 − 7y − 4y − 7 )
  • 5y[ ( y3 − 7y2 ) + ( − 4y − 7 ) ]
  • 5y[ y2( y − 7 ) − 4( y − 7 ) ]
  • 5y[ ( y2 − 4 )( y − 7 ) ]

5y[ ( y + 2 )( y − 2 )( y − 7 ) ]

Solve:
4x6 = 324x2

  • 4x6 − 324x2 = 0
  • 4x2( x4 − 81 ) = 0
  • 4x2( x2 + 9 )( x2 − 9 ) = 0
  • 4x2( x2 + 9 )( x + 3 )( x − 3 ) = 0
  • 4x2 = 0x = 0
  • x2 + 9 = 0x2 = − 9√{x2} = √{ − 9} (No Solution)
  • x + 3 = 0x = − 3
  • x − 3 = 0x = 3

x = { 0, − 3,3}

Factor:
64g2 − 80g + 25

( 8g − 5 )2

Factor:
121s2 + 154s + 49

( 11s + 7 )2

Factor:
81h2 − 180h + 100

( 9h − 10 )2

Factor:
20c3 − 60c + 45c

  • 5c( 4c2 − 12c + 9 )

5c( 2c − 3 )2

Factor:
50e3 − 120e2 + 72e

  • 2e( 25e2 − 60e + 36 )

2e( 5e − 6 )2

Factor:
192z3 + 576z2 + 432z

  • 3z( 64z2 + 192z + 144 )

3z( 8z + 12 )2

Solve:
2y2 − 16y + 32 = 50

  • 2( y2 − 8y + 16 ) = 50
  • 2( y − 4 )2 = 50
  • ( y − 4 )2 = 25
  • √{( y − 4 )2} = ±√{25} y − 4 = ±5 y − 4 = 5y = 9
  • y − 4 = − 5y = − 1

y{ − 1,9}

Solve:
3b2 + 150b + 75 = 147

  • 3( b2 + 50b + 25 ) = 147
  • 3( b + 5 )2 = 147
  • ( b + 5 )2 = 49
  • b + 5 = ±7
  • b + 5 = 7b = 2
  • b + 5 = − 7b = − 12

b = { − 12,2}

Solve:
16m2 − 48m + 36 = 256

  • 4(4m2 − 12m + 9) = 256
  • 4( 2m − 3 )2 = 256
  • ( 2m − 3 )2 = 64
  • 2m − 3 = ±8
  • 2m = ±11
  • 2m = 11m = [11/2]
  • 2m = − 11m = − [11/2]

m = { − [11/2],[11/2] }

Solve:
63a2 + 336a + 448 = 567

  • 7( 9a2 + 48a + 64 ) = 567
  • 7( 3a + 8 )2 = 567
  • ( 3a + 8 )2 = 81
  • 3a + 8 = ±9
  • 3a + 8 = 93a = 1a = [1/3]
  • 3a + 8 = − 93a = − 17a = − [17/3]

a = { − [17/3],[1/3] }

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Special Factoring Techniques

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Transcription: Special Factoring Techniques

Welcome back to www.educator.com.0000

In this lesson we are going take a look at some very special factoring techniques that you want to know.0002

Some of these special factoring techniques will be recognizing the difference of squares, 0009

a perfect square trinomial and the difference and sum of cubes.0015

Here is a great pattern to pick up on that can cut out a lot of work for you.0020

Recall that when we are multiplying together a few different polynomials that some very special ones came up. 0028

For example, if you multiply these two together x - y and x + y, the middle term end up dropping out entirely.0035

Our first term would be x2, our outside terms would be x, y, inside terms -xy and last terms –y2.0046

The only thing that will remain since these two cancel each other out are just the x2 and the - y2.0064

In few other situations you might have had, say x + y the whole thing squared and x - y the whole thing is squared of that one.0068

Since we have been working with factoring, we want to look at all 3 of these processes in the other way.0083

If we can recognize that we have one of these types of polynomials we will immediately know how it should factor on the other side.0090

Think of these as like some nice handy formulas that can help you cut down on the factoring process.0098

Since we are looking at these in reverse, we want to look at some key features of these formulas 0108

to help you recognize them when looking at those polynomials.0114

Let us start off.0117

When you are looking at the difference of squares here is how you can pick it out.0119

One, both of these terms should be squared and notice how there is no middle term.0124

One last very important thing is this is the difference of squares, so you should have subtraction.0138

You can find all those 3 parts then you know how this will end up factoring.0146

This will be x + y and x – y.0152

For your perfect square trinomials, you want the first term and the last term to be squared.0159

The last term over here should be positive. 0170

Now comes the tricky part, you want the middle term to be twice what you get when you multiply x and y together.0176

It does not matter if it is positive or negative, since you have two different formulas for those.0187

But that is what it should turn out to be. 0191

Now if you can recognize all of those things, then you know how these will break down.0195

This one will be x + y2 and that one will be x - y2.0199

Be careful when using these quick and easy formulas, note that there is no sum of squares formula.0207

It is often very tempting to look at something like x2 + y2 and think that it will break down, but be careful it will not.0216

Be very keen on picking up these small things that we can save yourself some time when going through that factoring process.0233

Let us give it a try.0244

I want to go ahead and factor x2 -162.0245

I think this looks like the difference of squares and I will end up rewriting it a little bit so you can tell.0250

This is x2 - and I need the other number to be2 as well, this would have to be a 42.0256

I'm looking at my two terms of being like an x and 4.0266

According to my formula, this will definitely factor.0271

Factor into x and x then +4 - 4.0277

Let us go ahead and multiply things back together to see why this is the correct factorization.0288

We take our first terms, we get x2, our outside would be - 4x, inside + 4x, and our last terms -16.0295

You will notice that those outside and inside terms, one is positive and one is negative, other than they are the same.0306

They will end up canceling each other out.0312

The only thing left with is just the x2 and the 16 like we should.0314

You can be pretty confident that this is the proper factorization when dealing with the difference of squares.0321

The other ones that we want to look at are those perfect square trinomials.0331

Let us see if we can hunt this one out. 0335

Looking at it, I can see that my y2 and I want to write the 100 as being a squared number as well.0337

y2 + 20y + the number 102.0344

My first term there is a y being squared and my last term is 10 being squared.0353

My last term is positive and that is a good sign. 0359

I need to check just one more thing, is my middle term twice the first term multiplied by the last term?0363

2 × y × 10.0372

It does not take too much work.0375

But if we do multiply these together or maybe rearrange them first, you will see that we do get a 20y like we are supposed to.0376

I can use my nice little shortcut formula to go ahead and factor this.0384

My first terms will be y and my last terms 10.0393

Notice how y + 10 and y + 10 they are exactly the same. 0403

We will go ahead and we write this as a single term with just a square exponent on it.0407

Nice, very quick and easy.0416

Let us go ahead and multiply this out to show you that it does work.0419

My first terms would be y2, my outside and inside terms would be exactly the same.0423

My last terms, 10 × 10 would be 100.0431

The fact that our outside and inside terms are exactly the same means that we will have two of them.0434

That is why we need that 2 in our shortcut formula.0440

That should make you pretty confident that this is the correct factorization for our perfect square trinomial.0444

Let us try some examples to see if we can get better using the special factoring techniques. 0452

In this first one I have y2 - 81and this looks like it might be the difference of squares.0457

I need to write this such that I have my first number squared and the second number squared.0467

92 is the only thing that will give me 1. 0474

I can definitely see that it is the difference of squares.0477

I have y2 92 and they are being subtracted.0480

This means that I can use my nice shortcut to go ahead and break this down. 0485

Y - 9 and y + 9.0490

Let us try something that has a few more fractions in it. 0500

This one is x2 - 16/25.0504

We want to write this such that we have something squared - something squared.0508

Two things that will multiply and give me x2, that will be x.0516

To get that 16/25 I think I we are going to have to use 4/5.0520

4 × 4 =16 and 5 × 5 =25.0526

Sure enough, this is the difference of squares because both of these are squared and we are taking their difference subtracted.0532

Let us write down what we have.0542

x - 4/5 and x + 4/5.0546

One more example, this one is u2 + 36.0555

You can recognize this one as u2 + 6 being squared. 0562

Be very careful, do not go anywhere beyond that because we cannot use one of our special formulas here.0568

Notice how this one we are dealing with a sum and we do not have a special formula for the sum of squares.0577

I’m going to write does not factor.0585

Be on the watch out for that one.0591

If you have the difference of squares you are in good shape. 0593

If you have the sum of squares then you are stocked.0595

Let us try a few more that involved the difference of squares. 0602

This first one we have a 49x2 - 25.0608

So we want to think of it as something squared - something squared.0613

Two numbers that will multiply and give me my 49, that must be 7.0620

7x is my first one.0626

Two numbers that will multiply to 25 must be 5.0629

We can write how this one breaks down.0637

I have 7x - 5 and 7x + 5.0641

One more, here I have two square numbers and we are taking their difference.0651

We need to view this as something squared - something squared.0658

See 8 × 8 = 64 I will put that in there. 0665

8p if I square that entire thing I will get 64p2.0671

9q, 9 × 9 will give us 81 and q2 will give us that q2.0678

Now we have both of our pieces and we can use our shortcut.0688

8p – 9q 8p + 9q and that one is factored completely.0696

When using any of the special factoring techniques you should always be on the watch out 0709

for some common factors that can pull out from the very beginning.0714

In these next two examples we will see if we can do just that.0718

This one 50 is not a square number nor is 32 but if I look at both of them, they are both divisible by 2.0722

I need to take out a common 2 first.0732

Let us try that common 2 and see what is left over.0736

5 0÷ 2 = 25w2 and 32 ÷ 2 = -16.0739

This is looking much better because 25 is a square number and so is 16. 0753

Let us just focus on those parts and see if we can write this as the difference of squares.0759

What would give us 25 must be a 52 and 16 that is a 42.0767

I can see how this one will factor.0777

5w - 4 and 5w + 4.0781

Do not forget that 2 that we took at the very beginning, it still hanging out front during the entire process. 0787

Go ahead and put it in.0795

One more of this difference of squares, but this one is a little bit of a change to it.0798

Notice how this one is actually to the 4th power. 0802

Sometimes you might be able to apply these rules, but you have to start off by looking out as something squared × something squared.0806

Let us see what we can do with this one.0817

What square would give us a y4?0819

This will be trickier but if you square a y2 that will do it.0824

It comes from our rules for exponents because we would end up multiplying the 2’s together.0829

What square would give us 81? That would be 9.0835

We could use our formula to break this down y2 - 9 and y2 + 9.0841

It is very tempting to stop right there but actually you can continue this one more step.0854

If you look over here you have another difference of squares.0859

You have a y2 and 9 can be factored out as 32.0863

Use the formula one more time to break that one down.0868

What square would give you a y2, y and what squared would give you 9 and 3.0874

This is y - 3 and y + 3. 0887

Now remember this one over on the other side is still there, go ahead and write it along with the rest.0893

Be careful not to try and apply your formulas to that one because that one is the sum of squares and we do not have a formula for that one.0900

Now that we have seen some of those, let us get into our perfect square trinomials.0912

The first one I have is x2 - 24x + 144.0917

Let us double check to see that this is a perfect square. 0922

My first term is squared and my last term is squared.0926

I'm looking at something like x2 - 24x and what will be squared to get 144, 122.0931

In order for this to work out nicely, I want to make sure that middle term comes from taking 2 multiplied by my first term x and last term 12.0944

By combining all that sure enough, you will see that we do get our 24x.0961

We are in pretty good shape.0966

This is -24 over here and that gives me another clue on how this will break down.0970

Break down into x - 12 and x - 12 which of course we just go ahead and package up into one x -12, the whole thing squared.0978

It is a very handy formula.0993

Let us try the next one.0995

18 x3 + 84x2 + 94.0997

That is quite a big one. 1002

In order to tackle this one, we definitely want a code for any common factors.1005

One thing I can say is one they are all divisible by 2 so that will help break down quite a bit.1011

And everything has x in common. 1016

Let us take those out and see what we have left over.1023

18 ÷ 2 = 9x2, 84 ÷ 2 = 42x and 98 ÷ 2 = 49.1030

Let us see if we can use our formula on this.1059

Out front I have a 9x2, on the back I have a 49.1062

You want to view this or at least that first one as being like a 3x the whole thing is being squared.1069

On the end that is like a 72.1075

Check to see that it meshes well with your middle term.1078

Can you take your first term 3x and your last term 7 and multiply by 2 to get 42.1082

2 × 3 × 7 =42 and there is still an x in there.1092

It matches just fine.1096

Let us use our formula and break that down.1098

That will be 3x + 7.1106

We are using the + in here because the 42 is positive 1115

and the reason why this is a 3x because that is what the first term squared would have to be.1120

Do not forget that 2x out front at the very beginning it is still there.1127

We will just finally go ahead and condense things since the 3x +7 appears twice.1131

I will write this 3x + 7 that whole thing squared with a 2x out front.1136

In addition to those special formulas that we have for factoring, there is also some for the sum and difference of cubes.1149

We have not done a lot of factoring with cubes so these are important in order to break these types of problems down.1158

For the sum and difference of cubes, they break down using these two formulas.1165

I think these are a little bit unusual because we look at what they factor into, they factor in some rather large polynomials.1170

You have this x - y, which is not too big, but then over here we have x2 + xy + y2.1178

Let me first convince you that it is how it should factor by taking those large polynomials and multiplying them together.1186

I’m going to do this using one of my tables. 1202

I will write the terms of one polynomial along the top and let me write the other one along the side. 1204

I’m using the first one here.1214

To fill in this box we will multiply x × x2 = x3, x2 × y = -x2y.1218

I have x2 y then - xy2 then xy2 then – ý3.1228

When I combine these terms I can see that all have a single x3, 1242

but that my x2 y will end up canceling each other out since they are different in sign.1251

The same thing will happen with my xy2, they are different in signs so they will cancel each other out as well.1257

I just have one little lonely y3 on the end, - ý3.1264

You can see that if I put these back together I do get the difference of cubes.1270

These two formulas will help us factor them out.1276

Another thing is it can be very difficult trying to remember these formulas.1281

First try and identify what is being cubed, this x and y, because you will see that they show up in your formula in that first polynomial.1286

Just x and y.1295

They show up in the second polynomial as well, x and y with them being multiplied in the middle.1297

It even works for the sum of cubes.1306

You have your x and y just as they are and you have your x, xy and y.1310

On the outside of the second one, these will always be squared.1317

And one last thing that will help you get these two formulas down is look at how the signs are related.1323

If you are dealing with the difference of squares in the first polynomial will have exactly the same sign.1330

If you have the sum of cubes in the first polynomial again will have the exact same sign.1337

The next sign present is opposite of what you used originally.1346

If you had a negative over here, use the opposite now it is positive.1351

Same thing applies on this one.1355

If you use the positive, now this was going to be negative.1357

For the last sign, this will always be positive. 1362

Here is how you can remember what the signs will be.1368

Always start off with the same one as in your original then you will have opposite signs.1371

Then the last one will always be positive.1380

Same, opposite, positive.1384

Let us see if we can use the sum and difference of cubes to help us factor up.1387

We are going to use this on 83 - 8 and 27r3 + 8.1394

The very first thing you do is see what two things are being cubed.1402

I have x and then 23 would give me my 8.1411

This will factor into a smaller polynomial and a larger one. 1419

The values that I put in here will be an x and 2, x2 x × 2 and 22.1427

I have put them the same as they are, then I have my x2.1440

I have them multiplied together and I have my 22.1444

We do cleanup this a little bit by putting some signs.1449

Same, opposite, positive.1454

This is almost a completely factored.1460

I will just go ahead and end up rewriting this because we usually like to put our coefficients in front of our variables.1464

2x rather than writing it as x times two and we should write this as 4 rather than 22.1474

Now that one is factored.1480

You can see they do take a little bit more work but it does get the job done.1482

Let us try this other one.1489

What cubed + what cubed.1492

What number cubed would give us 27?1497

That would have to be a 3.1502

We know that 3r is our first number in there.1504

What cubed could give us an 8?1507

I think we saw it before, that must have been 2.1509

This will breakdown into a smaller polynomial and a much larger one.1513

Let us write down our numbers.1519

3r and 2, 3r2 3r × 2 and 22. 1522

Let us go ahead and add the signs to this.1535

Same, opposite and positive.1538

Now this one is almost done. 1545

We just need to clean it up a little bit.1547

3r + 2 32 would be 9, 9r2 is here.1551

-3 × -2 = 6r and 22 would be 4.1559

This one is factored completely.1567

Let us do this one more time using the sum or the difference of cubes. 1574

Since it does take a little bit of practice to figure out what pieces are being cubed and where all of those pieces need to go. 1578

We need to recognize for this first one, something cubed + something cubed.1586

Looks like my first one, and let us see what cubed would give me 64?1596

That would have to be 4.1601

I’m thinking of breaking this down into one small piece and one larger piece.1605

We will go ahead and write down what these pieces are.1612

I have k, 4k2, k × 4 and 42.1614

Now that we have all of our pieces, let us go ahead and put in our signs.1626

Same, opposite, and positive. 1631

Do not forget this last step where we go ahead and clean everything up. 1639

k + 4k2 - 4k + 16.1644

Another thing that you may sometimes be tempted to do is sometimes you look at the second one 1655

and seems like you should be able to factor it in some sort of way.1662

However, this is as far as it goes.1666

Feel free to just leave it as it is.1669

Let us try one last one.1676

This one is 27x3 – 64y3.1677

Something cubed + something cubed.1683

What cubed would give us a 27?1691

That must be a 3 and x3 would give and x3.1694

To get a 64 this must be a 4y.1700

Make sure we have our negative sign in there.1706

We have that breakdown into a smaller one and a much larger one.1711

The pieces are 3x and 4y.1720

Be very careful as you put in those pieces of the much larger one.1724

Remember we have 3x2, we have 3x × 4y.1729

We have 4y all of that squared.1735

One common mistake I see with these is many people only square just the y.1738

But it is the entire thing that means to be squared.1744

The 4 and y.1746

I will put in some signs.1751

Same, opposite, and positive.1753

One last step, let us go ahead and clean it up.1761

3x – 4y now I have the entire thing 3x being squared.1765

That will be 9x2 3 × 4 would be 12, so 12 xy and 4y2 is 16y2.1773

This one is factored completely.1793

That definitely get familiar with the special formulas they can save you lots of time and works.1796

You do not have to go through as much.1801

Remember to look for those key patterns when using these formulas.1803

Always remember we do not have a sum of squares formula, so watch out for that one.1806

Thank you for watching www.educator.com.1812